On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces
The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity....
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2019/4358261 |
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| author | Moez Ben Abid Mourad Ben Slimane Ines Ben Omrane Borhen Halouani |
| author_facet | Moez Ben Abid Mourad Ben Slimane Ines Ben Omrane Borhen Halouani |
| author_sort | Moez Ben Abid |
| collection | DOAJ |
| description | The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism. |
| format | Article |
| id | doaj-art-d7685fd3b8724528860bacdb7d13d987 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-d7685fd3b8724528860bacdb7d13d9872025-02-03T06:12:16ZengWileyJournal of Function Spaces2314-88962314-88882019-01-01201910.1155/2019/43582614358261On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev SpacesMoez Ben Abid0Mourad Ben Slimane1Ines Ben Omrane2Borhen Halouani3Université de Sousse, Ecole Supérieure des Sciences et de la Technologie de Hammam Sousse, Sousse, TunisiaKing Saud University, Department of Mathematics, College of Science, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi ArabiaKing Saud University, Department of Mathematics, College of Science, P.O. Box 2455, Riyadh 11451, Saudi ArabiaThe t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.http://dx.doi.org/10.1155/2019/4358261 |
| spellingShingle | Moez Ben Abid Mourad Ben Slimane Ines Ben Omrane Borhen Halouani On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces Journal of Function Spaces |
| title | On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces |
| title_full | On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces |
| title_fullStr | On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces |
| title_full_unstemmed | On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces |
| title_short | On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces |
| title_sort | on the baire generic validity of the t multifractal formalism in besov and sobolev spaces |
| url | http://dx.doi.org/10.1155/2019/4358261 |
| work_keys_str_mv | AT moezbenabid onthebairegenericvalidityofthetmultifractalformalisminbesovandsobolevspaces AT mouradbenslimane onthebairegenericvalidityofthetmultifractalformalisminbesovandsobolevspaces AT inesbenomrane onthebairegenericvalidityofthetmultifractalformalisminbesovandsobolevspaces AT borhenhalouani onthebairegenericvalidityofthetmultifractalformalisminbesovandsobolevspaces |